96 Some basic maths for seismic data processing and inverse problems

Some basic maths for seismic data processing and inverse problems (Refreshement only!)  Complex Numbers  Vectors  Linear vector spaces  Linear systems  Matrices     Determinants Eigenvalue problems Singular values Matrix inversion Mathematical foundations  Series  Taylor  Fourier  Delta Function  Fourier integrals The idea is to illustrate these mathematical tools with examples from seismology Complex numbers iφ z = a + ib = re = r (cos φ + i sin φ ) Mathematical foundations Complex numbers conjugate, etc z* = a − ib = r (cos φ − i sin φ ) = r cos − φ − ri sin( −φ ) = r −iφ z = zz* = ( a + ib)(a − ib) = r cos φ = (e iφ + e −iφ ) / sin φ = (e iφ − e −iφ ) / 2i Mathematical foundations Complex numbers seismological applications     Discretizing signals, description with eiwt Poles and zeros for filter descriptions Elastic plane waves Analysis of numerical approximations ui ( x j , t ) = Ai exp[ik (a j x j − ct )] u(x, t ) = A exp[ikx − ωt ] Mathematical foundations Vectors and Matrices For discrete linear inverse problems we will need the concept of linear vector spaces The generalization of the concept of size of a vecto to matrices and function will be extremely useful for inverse problems Definition: Linear Vector Space A linear vector space over a field F of scalars is a set of elements V together with a function called addition from VxV into V and a function called scalar multiplication from FxV into V satisfying the following conditions for all x,y,z ∈ V and all a,b ∈ F (x+y)+z = x+(y+z) x+y = y+x There is an element in V such that x+0=x for all x ∈ V For each x ∈ V there is an element -x ∈ V such that x+(-x)=0 a(x+y)= a x+ a y (a + b )x= a x+ bx a(b x)= ab x 1x=x Mathematical foundations Matrix Algebra – Linear Systems Linear system of algebraic equations a11 x1 + a12 x2 + + a1n xn = b1 a21 x1 + a22 x2 + + a2 n xn = b2 an1 x1 + an x2 + + ann xn = bn where the x1, x2, , xn are the unknowns in matrix form Ax = b Mathematical foundations Matrix Algebra – Linear Systems Ax = b  a11 a 21  A = aij =   an1 [ ] where  a1n   a22  a11      an  ann  a12  b1  b   2 b = { bi } =     bn  Mathematical foundations  x1  x   2 x = { xi } =      xn  A is a nxn (square) matrix, and x and b are column vectors of dimension n Matrix Algebra – Vectors Row vectors v= [ v1 v2 Column vectors w    w=  w2  w   3 v3 ] Matrix addition and subtraction C = A +B D = A −B with with cij = aij + bij d ij = aij − bij Matrix multiplication C = AB with m cij = ∑ aik bkj k =1 where A (size lxm) and B (size mxn) and i=1,2, ,l and j=1,2, ,n Note that in general AB≠BA but (AB)C=A(BC) Mathematical foundations Matrix Algebra – Special Transpose of a matrix [ ] A = aij Symmetric matrix [ ] A T = a ji ( AB) T = BT A T Identity matrix 1  0 0  0  I=       0  1 with AI=A, Ix=x Mathematical foundations A = AT aij = a ji Matrix Algebra – Orthogonal Orthogonal matrices a matrix is Q (nxn) is said to be orthogonal if QT Q = I n and each column is an orthonormal vector qi qi = examples: Q= 1 − 1 1    it is easy to show that : QT Q = QQT = I n if orthogonal matrices operate on vectors their size (the result of their inner product x.x) does not change -> Rotation (Qx)T (Qx) = xT x Mathematical foundations Eigenvalue problems … one of the most important tools in stress, deformation and wave problems! It is a simple geometrical question: find me the directions in which a square matrix does not change the orientation of a vector … and find me the scaling … Ax = λx the rest on the board … Mathematical foundations Some operations on vector fields Gradient of a vector field  ∂x   ∂ x  ux   ∂ x ux       ∇u =  ∂ y u =  ∂ y  u y  =  ∂ x u y ∂   ∂  u   ∂ u  z  z  z   x z ∂ yux ∂ yu y ∂ yu z What is the meaning of the gradient? Mathematical foundations ∂ z ux   ∂ zuy   ∂ z uz  Some operations on vector fields Divergence of a vector field  ∂x   ∂ x   ux        ∇ • u =  ∂ y  • u =  ∂ y  •  u y  = ∂ x u x + ∂ yu y + ∂ z u z ∂  ∂  u   z  z  z When u is the displacement what is ist divergence? Mathematical foundations Some operations on vector fields Curl of a vector field  ∂x   ∂ x   u x   ∂ yu z − ∂ z u y          ∇ × u =  ∂ y  × u =  ∂ y  ×  uy  =  ∂ zux − ∂ x uz  ∂  ∂  u  ∂ u − ∂ u  y x  z  z  z  x y Can we observe it? Mathematical foundations Vector product A = a × b = a b sin θ Mathematical foundations Matrices –Systems of equations Seismological applications  Stress and strain tensors  Calculating interpolation or differential operators for finite-difference methods  Eigenvectors and eigenvalues for deformation and stress problems (e.g boreholes)  Norm: how to compare data with theory  Matrix inversion: solving for tomographic images  Measuring strain and rotations Mathematical foundations The power of series Many (mildly or wildly nonlinear) physical systems are transformed to linear systems by using Taylor series 1 f ( x + dx) = f ( x) + f ' dx + f ' ' dx + f ' ' ' dx + ∞ f (i ) ( x) i =∑ dx i! i =1 Mathematical foundations … and Fourier Let alone the power of Fourier series assuming a periodic function … (here: symmetric, zero at both ends) f ( x ) = a0 + ∑ n n   an sin  2πx  2L   L a0 = ∫ f ( x)dx L0 nπx an = ∫ f ( x) sin dx L0 L L Mathematical foundations n = 1, ∞ Series –Taylor and Fourier Seismological applications  Well: any Fouriertransformation, filtering  Approximating source input functions (e.g., step functions)  Numerical operators (“Taylor operators”)  Solutions to wave equations  Linearization of strain - deformation Mathematical foundations The Delta function … so weird but so useful … ∫ ∞ ∫ ∞ −∞ −∞ δ (t ) f (t )dt = f (0) δ (t )dt = , δ (t ) = f (t )δ (t − a ) = f (a ) δ (at ) = δ (t ) a δ (t ) = Mathematical foundations 2π ∞ i ωt e ∫ dω −∞ für t ≠ Delta function – generating series Mathematical foundations The delta function Seismological applications  As input to any system (the Earth, a seismometers …)  As description for seismic source signals in time and space, e.g., with Mij the source moment tensor s (x, t ) = Mδ (t − t0 )δ (x − x )  As input to any linear system -> response Function, Green’s function Mathematical foundations Fourier Integrals The basis for the spectral analysis (described in the continuous world) is the transform pair: f (t ) = 2π F (ω ) = ∞ ∫ ∞ i ωt F ( ω ) e dω ∫ −∞ f (t )eiωt dt −∞ For actual data analysis it is the discrete version that plays the most important role Mathematical foundations Complex fourier spectrum The complex spectrum can be described as F (ω ) = R(ω ) + iI (ω ) = A(ω )e iΦ (ω ) … here A is the amplitude spectrum and Φ is the phase spectrum Mathematical foundations The Fourier transform Seismological applications • Any filtering … low-, high-, bandpass • Generation of random media • Data analysis for periodic contributions • Tidal forcing • Earth’s rotation • Electromagnetic noise • Day-night variations • Pseudospectral methods for function approximation and derivatives Mathematical foundations ... applications  Stress and strain tensors  Calculating interpolation or differential operators for finite-difference methods  Eigenvectors and eigenvalues for deformation and stress problems (e.g boreholes)... spectrum and Φ is the phase spectrum Mathematical foundations The Fourier transform Seismological applications • Any filtering … low-, high-, bandpass • Generation of random media • Data analysis for. .. foundations Vectors and Matrices For discrete linear inverse problems we will need the concept of linear vector spaces The generalization of the concept of size of a vecto to matrices and function will